ImageMagick

set im_convert="C:\Program Files\ImageMagick-6.5.8-Q16\convert.exe"
set input_image=post_processed.tif
set output_image=nadir_balled.tif
rem Make the mirror ball.
rem -crop takes the part of the image without the nadir
rem -resize resizes the image to height of the nadir
rem -flip flips the image vertically
rem -level darkens the image, because reflection is never 100 percent
%im_convert% %input_image% -crop 12780x6130+0+0 -resize 12780x260! -flip -level 0,130%% temp_nadir_ball.tif
rem Make a gradient image from black to transparent. This is the ambient shadow that is casted by the mirror ball.
%im_convert% -size 12780x130 gradient:none-black temp_gradient.png
rem Place the shadow and the mirror ball on the image
%im_convert% %input_image% -page +0+6070 temp_gradient.png -flatten -page +0+6130 temp_nadir_ball.tif -flatten %output_image%
del temp_nadir_ball.tif
del temp_gradient.png
pause

Before using it, please adapt the following in the batch file to your situation:

the location of ImageMagick's convert.exe

the input panorama image name

the output panorama image name

the size of the panorama without the nadir cap (12780x6130)

the size of the nadir cap (12780x260)

the size of the ambient shadow (12780x130)

the position of the ambient shadow on the output panorama image (+0+6070)